Question Number 93827 by oustmuchiya@gmail.com last updated on 15/May/20
$${simplify}:\:\frac{\left({cos}\alpha+\boldsymbol{\mathrm{i}}{sin}\alpha\right)^{\mathrm{3}} }{\left({sin}\beta+\boldsymbol{\mathrm{i}}{cos}\beta\right)^{\mathrm{4}} } \\ $$
Commented by PRITHWISH SEN 2 last updated on 15/May/20
$$\frac{\mathrm{e}^{\boldsymbol{\mathrm{i}}\mathrm{3}\boldsymbol{\alpha}} }{\mathrm{e}^{\boldsymbol{\mathrm{i}}\mathrm{4}\left(\frac{\pi}{\mathrm{2}}−\beta\right)} }\:=\:\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{i}}\left(\mathrm{3}\boldsymbol{\alpha}+\mathrm{4}\boldsymbol{\beta}\right)} \:=\:\boldsymbol{\mathrm{cos}}\left(\mathrm{3}\boldsymbol{\alpha}+\mathrm{4}\boldsymbol{\beta}\right)+\boldsymbol{\mathrm{isin}}\left(\mathrm{3}\boldsymbol{\alpha}+\mathrm{4}\boldsymbol{\beta}\right) \\ $$